Proving a convex function is of bounded variation

Question

Let $f: I \rightarrow \mathbb{R}$ be a convex function, where $I$ denotes a closed interval. If $f$ is convex, how would I prove that $f$ is of bounded variation?

Answer 1

By convexity there cannot exist $a<b<c \in I$ such that $f(a)<f(b)$ and $f(c)\leq f(b)$ (draw a picture). So if $a_0<a_1<\ldots <a_n$ is a partition of $I$ we might start off by having $f(a_0)>f(a_1)$, but the moment we have a $k$ such that $f(a_k)<f(a_{k+1})$ we must have $f(a_i)<f(a_{i+1})$ for all $i\geq k$. Then $$\sum\limits_{i=0}^{n-1} |f(a_{i+1})-f(a_i)|=\sum\limits_{i=0}^{k-1} f(a_i)-f(a_{i+1})+ \sum\limits_{i=k}^{n-1} f(a_{i+1})-f(a_i)=f(a_0)-2f(a_k)+f(a_n),$$

and so we only have to show that $f$ is bounded on the closed interval $I$ and we will be done. For this, write $I=[x, z]$ and let $y$ be any point strictly in between $x$ and $z$. Let $f_1$ be the linear interpolation of the points $(x,f(x))$ and $(y, f(y))$, and let $f_2$ be the linear interpolation of the points $(y, f(y))$ and $(z, f(z))$. We then see that for $a\in [x,y]$, $f_2(a)\leq f(a) \leq f_1(a)$ and for $a\in [y,z]$, $f_1(a)\leq f(a) \leq f_2(a)$, so $f$ is bounded on $I$.

Answer 2

A small variation (!) of Jonathan's answer: Let $f: [a, b] \to \Bbb R$ be convex. Let $$ c = \sup \{ x \in [a, b] \mid \text{$f$ is decreasing on $[a, x]$} \} \, . $$ Then $f$ is decreasing on $[a, c]$ and increasing on $[c, b]$, and therefore $$ V_a^b(f) = V_a^c(f) + V_c^b(f) = (f(a)-f(c)) + (f(b) - f(c)) = f(a) - 2f(c) + f(b) < \infty \, . $$

In other words: a convex function is piecewise monotone and therefore of bounded variation on a compact interval.